PM brushless motor drive circuit topology and control

ABSTRACT

An inverter for a permanent magnet brushless dc machine, having a permanent magnet rotor and a set of stator windings, applies the full dc voltage provided to the inverter to each phase of the machine.

This application claims priority to U.S. provisional application61/318,506 filed Mar. 29, 2010, the content of which is incorporatedherein by reference.

FIELD OF THE INVENTION

The invention relates to an inverter controlled permanent magnetbrushless direct current (PMBDC) motor drive system. More specifically,the invention relates to the realization of a new inverter and itscontrol in conjunction with a PMBDC machine to improve the operation ofthe entire motor drive system.

BACKGROUND OF THE RELATED ART

Permanent magnet brushless direct current (PMBDC) machines are wellknown in literature. Consider a three phase machine having three phasewindings and a rotor with permanent magnets. The phase windings arebalanced, in that they have the same number of turns per phase winding,and they are spatially phase shifted by 120 electrical degrees. Thetheory and operation of such machines are described in chapters 1 and 9of the book, R. Krishnan, “Permanent Magnet Synchronous and Brushless DCMotor Drives”, CRC Press, 2009. The theory and operation of a threephase H-bridge inverter is common knowledge and described in the samebook in chapters 2 and 10.

Consider the magnitude of a direct current (dc) input voltage to aninverter as V_(s1), which can be supplied from either a battery sourceor a rectified alternating current (ac) source. Assuming two phasewindings of a machine are series connected, the instantaneous maximumvoltage that is applied across a machine winding phase through a threephase inverter is 0.5V_(s1). If the current in a phase is 1, then theinput power per phase is 0.5V_(s1)I. Since two phases conduct at anygiven time in a PMBDC motor drive, the maximum input power to the PMBDCmachine is 2*0.5V_(s1)I=V_(s1)I.

Because the two windings are in series, they carry the same current I.From the dc link or input side, the instantaneous power supplied to theinverter and machine is the product of voltage V_(s1) and current I,expressed as V_(s1)I. The instantaneous power is equal to the inverterinput power and the machine input power, if losses in the inverter areignored.

The inverter supplies each phase with 120 electrical-degrees-widecurrent in both positive and negative cycles; that is, the invertersupplies a bipolar or alternating current to the machine phase windings.Furthermore, an inverter phase leg has an ideal duty cycle of ⅔, meaningthat it is active for that part of the time in the machine's ac cycle.For the remaining ⅓ of the cycle time, the inverter is in the offcondition (i.e., it is not active). This is true for all three phases inthe inverter.

The machine phase windings carry the alternating currents generatedthrough the inverter, though the current of each winding is phaseshifted from the others by 120 electrical degrees. The phase shift ofthe current in each winding is the same amount as the spatial phaseshift between the winding phases, so as to produce a uniform andconstant air gap torque and power.

Consider the winding resistive losses in the PMBDC machine and let theresistance per phase be R_(s) in units of Ohms. The total instantaneouswinding power loss is equal to I²R_(s) per phase. Since two windingphases conduct in a PMBDC machine at any given time, the total resistiveloss is 2I²R_(s).

SUMMARY OF THE INVENTION

Because the resistive loss of a permanent magnet brushless directcurrent (PMBDC) machine is 2I²R_(s), the voltage applied to a phase maybe increased without affecting the resistive loss. As the voltageapplied to a phase is increased, the phase current may becorrespondingly decreased without affecting the input power to themachine. However, the resistive loss decreases as the phase currentdecreases, resulting in higher operational efficiency of the motor. Animproved inverter, for driving a PMBDC machine, that applies highervoltage and lower current to a phase, so as to reduce resistive loss, isan object of this invention.

Other objects of the invention include increasing the applied voltageacross a machine phase to greater than half a direct current (dc) inputvoltage, decreasing conductive losses in inverter transistors,decreasing combined power losses, achieving high efficiency of a motordrive system, achieving higher speed of operation for constant torqueoperation, achieving alternating current and voltage control with higherphase voltages in the machine, achieving four quadrant operation, andincreasing reliability and fault tolerance for the drive system. Acontrol system to coordinate the operation of the motor drive system isa further objective of the invention. To achieve these objects, theinverter disclosed hereinafter produces an input voltage that is greaterthan or comparable to that of a dc input supply voltage.

These and other objects of the invention may be achieved by a motordrive system having: (1) a first capacitive element that stores energyso as to provide a first dc voltage supply and (2) a second capacitiveelement that stores energy so as to provide a second dc voltage supply.A first electrically conductive switch electrically connects in serieswith a winding of a single motor phase and the first capacitive elementto form a first series circuit when the first switch conducts current. Asecond electrically conductive switch electrically connects in serieswith the motor phase winding and the second capacitive element to form asecond series circuit when the second switch conducts current. Theentire voltage of the first capacitive element is applied across themotor phase winding, less the voltage drop across the first switch andthe voltage drop resulting from parasitic resistance in the first seriescircuit, when: (1) the second switch does not conduct current and (2)the second capacitive element is neither being charged nor discharged.The entire voltage of the second capacitive element is applied acrossthe motor phase winding, less the voltage drop across the second switchand the voltage drop resulting from parasitic resistance in the secondseries circuit, when: (1) the first switch does not conduct current and(2) the first capacitive element is neither being charged nordischarged.

Moreover, the above-mentioned and other objects of the invention may beachieved by a motor drive system having: (1) a phase winding of a motorthat applies electromotive force to a rotor of the motor; (2) a firstcapacitive element that stores energy so as to provide a first dcvoltage supply; and (3) a second capacitive element that stores energyso as to provide a second dc voltage supply. A first electricallyconductive switch electrically connects in series with the motor phasewinding and the first capacitive element to form a first series circuitwhen the first switch conducts current. A second electrically conductiveswitch electrically connects in series with the motor phase winding andthe second capacitive element to form a second series circuit when thesecond switch conducts current. The entire voltage of the firstcapacitive element is applied across the motor phase winding, less theVoltage drop across the first switch and the voltage drop resulting fromparasitic resistance in the first series circuit, when: (1) the secondswitch does not conduct current and (2) the second capacitive element isneither being charged nor discharged. The entire voltage of the secondcapacitive element is applied across the motor phase winding, less thevoltage drop across the second switch and the voltage drop resultingfrom parasitic resistance in the second series circuit, when: (1) thefirst switch does not conduct current and (2) the first capacitiveelement is neither being charged nor discharged.

Furthermore, the above-mentioned and other objects of the invention maybe achieved by a motor drive system having a first capacitive elementthat stores energy so as to provide a first dc voltage supply, a secondcapacitive element that stores energy so as to provide a second dcvoltage supply, and first, second, third, and fourth electricallyconductive switches each having an input current terminal and an outputcurrent terminal. A first terminal of the first capacitive element iselectrically connected directly to the input current terminals of thefirst and second switches. A second terminal of the first capacitiveelement is electrically connected directly to a first terminal of thesecond capacitive element. A second terminal of the second capacitiveelement is electrically connected directly to the output currentterminals of the third and fourth switches. The output current terminalof the first switch is electrically connected directly to the inputcurrent terminal of the third switch. The output current terminal of thesecond switch is electrically connected directly to the input currentterminal of the fourth switch. A first terminal of a winding of a firstmotor phase is electrically connected directly to the second terminal ofthe first capacitive element and the first terminal of the secondcapacitive element. A second terminal of the first motor phase windingis electrically connected directly to the output current terminal of thefirst switch and the input current terminal of the third switch. A firstterminal of a winding of a second motor phase is electrically connecteddirectly to the second terminal of the first capacitive element and thefirst terminal of the second capacitive element. A second terminal ofthe second motor phase winding is electrically connected directly to theoutput current terminal of the second switch and the input currentterminal of the fourth switch.

Still further, the above-mentioned and other objects of the inventionmay be achieved by a motor drive method that discharges energy stored bya first capacitive element into a single electrical phase of a motor,via a first electrically conductive switch, such that the entire voltageacross the first capacitive element is applied across the motor phasewinding, less the voltage drop across the first switch and the voltagedrop resulting from parasitic resistance in a first series circuitcomprising the first capacitive element, the first switch, and the motorphase winding. Additionally, the energy stored by the motor phasewinding is discharged into a second capacitive element, via a secondelectrically conductive switch, such that the entire voltage across themotor phase winding is applied across the second capacitive element,less the voltage drop across the second switch and the voltage dropresulting from parasitic resistance in a second series circuitcomprising the second capacitive element, the second switch, and themotor phase winding.

In accordance with the exemplary embodiment of the invention, a PMBDCmachine drive system has a machine with permanent magnet (PM) rotor anda stator with three phase windings. The three phase stator windings arespatially shifted from each other by 120 electrical degrees and have aneutral connection amongst them. The induced electromotive forces (emfs)across machine phases are, ideally, trapezoidal waveforms with aconstant region over 120 electrical degrees in both half cycles; adeviation from such ideality has no impact on the invention andobjectives.

The PMBDC system has an H-bridge inverter to supply currents to thestator phase windings. The inverter's input dc voltage positive polarityis connected to the top rail of the inverter, and the input voltage'snegative polarity is connected to the neutral of the machine statorphase windings. A dc link capacitor C₁ is connected between the positiveand negative polarities of the input dc voltage. An additional dccapacitor C₂ is connected between the neutral and bottom rail of thethree phase inverter subsystem.

A supply dc input voltage V_(s1), operating in conjunction with themachine windings and diodes in the bottom half of the inverter bridge,charges capacitor C₂, thereby generating a voltage across it. Thecharging of capacitor C₂ happens whenever conducting transistors in thetop half of the inverter bridge are turned off and the current in themachine phase operates to forward bias the diodes in the bottom inverterhalf. The terminal of capacitor C₂ tied to the neutral side of themachine side is charged to a positive polarity and the terminal ofcapacitor C₂ tied to the bottom rail side of the inverter is charged tonegative polarity.

With this arrangement, the machine phase windings have available, ontheir positive cycle operation, a voltage equal to the dc input supplyvoltage V_(s1), through the control of the transistors in the top halfof the inverter bridge. On the negative cycle operation of the machine,phase windings are supplied a voltage equal to the voltage availableacross C₂, i.e., V_(s2), through the operation of transistors in thebottom half of the inverter bridge.

When the conducting top transistor of a phase leg is turned off, thecurrent in the phase corresponding to this transistor is taken over bythe diode in the bottom of the phase leg, completing the current pathvia capacitor C₂. Likewise, when the conducting bottom transistor of aphase leg is turned off, the current in the phase winding is routedthrough the top diode of the phase leg, completing the current path viacapacitor C₁ and charging it. This operation partially transfers theenergy from the machine phase winding to a capacitor, and part of theenergy is converted to power input for the machine. The voltages acrosscapacitors C₁ and C₂ may be the same or different.

The drive system may use a standard H bridge inverter and a neutralconnection of a three-phase machine, so as not to add additional costbeyond that of a conventional system. However, the drive system requiresan additional capacitor C₂ and higher voltage ratings for transistorsand diodes in the inverter, which do add an incremental cost to thetotal PMBDC motor drive system.

The drive system inverter and PMBDC motor disclosed herein reduce theconduction losses in the inverter transistors and diodes. The lowerlosses in the inverter cause lower total losses, resulting in higherefficiency. Further, the drive system provides higher peak poweroperation and higher speeds of operation at full load torque.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a related art inverter permanent magnet brushlessdirect current (PMBDC) motor drive system;

FIG. 2 illustrates the operation of the PMBDC motor drive system of FIG.1;

FIG. 3 illustrates an inverter PMBDC motor drive system;

FIG. 4 illustrates the operation of the PMBDC motor drive system of FIG.3;

FIG. 5 illustrates the four quadrants of torque vs. speed and theoperational domain available to the PMBDC motor drive system of FIG. 3;

FIG. 6 illustrates the unidirectional control operation of the PMBDCmotor drive system of FIG. 3;

FIG. 7 illustrates the duty cycle of a positive-current conducting phasevs. current magnitude as a function of normalized stator phaseresistance; and

FIG. 8 illustrates an uninterruptible-supply power converter.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 illustrates an inverter permanent magnet brushless direct current(PMBDC) motor drive system of the related art. A direct current (dc)supply voltage 100, obtained either from a rectifier connected to theutility supply or from a battery, is provided to an inverter 110. Afiltering capacitor C₁ is connected in parallel with dc supply voltage100, via a upper voltage rail 101 and a lower voltage rail 102, tosmooth the voltage variations of dc supply voltage 100. The dc voltageexisting across capacitor C₁ and provided to inverter 110 is expressedas V_(s1).

Inverter 110 is a three-phase H bridge inverter and is electricallyconnected in parallel with capacitor C₁ via voltage rails 101 and 102.Inverter 110 has two transistors in each of three phase legs. Phase legA includes transistors T₁ and T₄ electrically connected in series. Adiode D₁ is connected in parallel with transistor T₁, such that thecathode of diode D₁ is electrically connected to transistor T₁'scollector and the anode of diode D₁ is electrically connected to theemitter of transistor T₁. A diode D₄ is similarly connected in parallelwith transistor T₄. Phase leg B includes transistors T₃ and T₆electrically connected in series, and phase leg C includes transistorsT₅ and T₂ electrically connected in series. Each of transistors T₂, T₃,T₅, and T₆ has a corresponding diode D₂, D₃, D₅, and D₆ electricallyconnected in parallel with the respective transistor, in the same waythat diode D₁ is connected to transistor T₁.

The current conduction in each of transistors T₁-T₆ and diodes D₁-D₆ isindicated by an arrow in FIG. 1. Each of transistors T₁-T₆ conductscurrent from collector to emitter, and each of diodes D₁-D₆ conductscurrent from anode to cathode.

The collectors of transistors T₁, T₃, and T₅ and cathodes of diodes D₁,D₃, and D₅ are electrically connected to upper voltage rail 101. Theemitters of transistors T₂, T₄, and T₆ and anodes of diodes D₂, D₄, andD₆ are electrically connected to lower voltage rail 102. The gates oftransistors T₁-T₆ are individually controlled to govern the flow ofcurrent through the respective transistor.

A PMBDC motor 105 has three phase stator windings with phases A, B, andC, Phase leg A controls the current flow through the windings of phaseA, phase leg B controls the current flow through the windings of phaseB, and phase leg C controls the current flow through the windings ofphase C. One end of the stator winding for phase A is electricallyconnected to neutral N and the other end is electrically connected tothe emitter of transistor T₁, the collector of transistor T₄, the anodeof diode D₁, and the cathode of diode D₄. Similarly, one end of thestator winding for phase B is electrically connected to neutral N andthe other end is electrically connected to the emitter of transistor T₃,the collector of transistor T₆, the anode of diode D₃, and the cathodeof diode D₆. And similarly, one end of the stator winding for phase C iselectrically connected to neutral N and the other end is electricallyconnected to the emitter of transistor T₅, the collector of transistorT₂, the anode of diode D₅, and the cathode of diode D₂.

The theory and operation of the PMBDC machine illustrated in FIG. 1 aredescribed in chapters 1 and 9 of the book, R. Krishnan, “PermanentMagnet Synchronous and Brushless DC Motor Drives”, CRC Press, 2009. Thetheory and operation of the three phase H-bridge inverter illustrated inFIG. 1 are described in the same book, in chapters 2 and 10.

FIG. 2 illustrates the operation of the PMBDC motor drive system ofFIG. 1. The induced electromotive forces (emfs) of phases A, B, and Care denoted as e_(as), e_(bs) and e_(cs), respectively. The induced emfof a phase is the instantaneous voltage induced across its correspondingphase winding. Each of phases A-C of the PMBDC machine has atrapezoidal, induced emf with a constant voltage magnitude of E_(p),both in the phase's positive and negative 120° cycles, for some periodof a 360° cycle and a ramp voltage rising from −E_(p) to +E_(p) orfalling from +E_(p) to −E_(p) during the remaining period of the 120°cycle. The induced emfs of each of phases A-C are electrically phaseshifted from each other by 120°. The induced emf magnitude E_(p) isexpressed as:E _(p)(Blv)N=N(Blrω _(m))=Nφ _(a)ω_(m)=λ_(p)ω_(m)  (1)

where N is the number of conductors in series per phase, v is thevelocity, l is the length of the conductor, r is the radius of the rotorbore, ω_(m) is the angular velocity, and B is the flux density of thefield in which the conductors are placed. This flux density is solelydue to the rotor magnets. The product (Blv), also expressed as φ_(a),has the dimensions of flux and is directly proportional to the airgapflux φ_(g) expressed in equation (2):

$\begin{matrix}{\phi_{a} = {{Blr} = {{\frac{1}{\pi}B\;{\pi lr}} = {\frac{1}{\pi}\phi_{g}}}}} & (2)\end{matrix}$

The product of flux and number of conductors in series has the dimensionof flux linkage and is denoted by λ_(p). Since the product of flux andnumber of conductors in series is proportional to phase flux linkage bya factor of

$\frac{1}{\pi},$the product is hereafter referred to as modified flux linkage.

The electromagnetic torque is given by:

$\begin{matrix}{{T_{e} = {\lbrack {{e_{as}i_{as}} + {e_{bs}i_{bs}} + {e_{cs}i_{cs}}} \rbrack\frac{1}{\omega_{m}}}},\mspace{14mu}{N \cdot m}} & (3)\end{matrix}$

The instantaneous induced emfs, having units of volts, may be expressedas:e _(as) =f _(as)(θ_(r))λ_(p)ω_(m)  (4)e _(bs) =f _(bs)(θ_(r))λ_(p)ω_(m)  (5)e _(cs) =f _(cs)(θ_(r))λ_(p)ω_(m)  (6)

where the functions f_(as)(θ_(r)), f _(bs)(θ_(r)) and f _(cs)(θ_(r))have the same shape as e_(as), e_(bs) and e_(cs), with a maximummagnitude of ±1.

The electromagnetic torque, having units of N·m, may be expressed, aftersubstituting for induced emfs from equations (4, 5 and 6) into equation(3), as:T _(e)=λ_(p) [f _(as)(θ_(r))i _(as) +f _(bs)(θ_(r))i _(bs) +f_(cs)(θ_(r))i _(cs) ],N·m  (7)

Currents are generated when the functions f_(as)(θ_(r)), f_(bs)(θ_(r))and f_(cs)(θ_(r)) attain a value of ±1 in PMBDC motor drives. Only twoof the three phases are excited, with current going into one (withpositive function value of 1, say for phase A) and the current comingout of the other one of the excited phases (with the negative functionvalue of −1, say for phase B), such that the electromagnetic torquebecomes:T _(e)=λ_(p) [i _(as)+(−1)i _(bs) ],N·m.  (8)

For an example in which phases A and B are excited, if the current inphase A is positive, then the current in phase B is negative. Thecurrents in phases A and B are equal in magnitude because they areproduced by turning on transistors T₁ and T₆, and the activation oftransistors T₁ and T₆ causes phase A winding and phase B winding to beconnected in series against dc supply voltage 100. Denoting the currentin both phases as I_(p), the torque equation may be expressed as:T _(e)=2λ_(p) I _(p) ,N·m.  (9)

The air gap power acquired from summing the products of the induced emfsof the phases, with their respective currents, at any time is expressedby the equation:P _(o) =└e _(as) i _(as) +e _(bs) i _(bs) +e _(cs) i _(cs) ┘=E _(p) I_(p)+(−E _(p))(−I _(p))=2E _(p) I _(p)  (10)

FIG. 2 illustrates the phase-induced emfs, respective phase currents,respective air gap power for each phase, and total air gap power for themachine as a function of electrical rotor position.

Suppose the related art is maintained the same, as shown in FIG. 2,except that the induced emf is increased by a factor of k for all phasesand the applied currents from the inverter is reduced by a factor of 1/kfor all phase currents. The induced emfs are increased by increasing thenumber of turns in the phases. The operation with such scaling isillustrated in FIG. 4, which is very similar to FIG. 2, and all thevariables and their symbols mean the same things in both figures. Theconsequence of such an operation yields the torque and air gap power as:

$\begin{matrix}{P_{o} = {{{{kE}_{p}\frac{I_{p}}{k}} + {( {- {kE}_{p}} )( {- \frac{I_{p}}{k}} )}} = {2E_{p}I_{p}}}} & (11) \\{{T_{e} = {2\lambda_{p}I_{p}}},\mspace{14mu}{N.m.}} & (12)\end{matrix}$

Nothing has changed in the air gap power or in the electromagnetictorque by increasing the induced emf and decreasing by the sameproportion the applied currents to the machine phases between the scaledapproach and the related art system. The induced emf for the scaledapproach may be increased to match that the supply voltage, and thephase currents can be decreased by the same proportion to which theinduced emfs are increased.

Usually the alternating current (ac) or dc supply voltage is a fixedvalue and not subject to change in practical applications. However, thedc supply voltage to the inverter may be increased by boosting therectified ac voltage or battery voltage by a boost converter placedbetween the dc supply and the inverter. This modification introduces anadditional power converter, which decreases system efficiency andincreases the number of power electronic components and system cost.

Thus, it would be desirable to make the same battery voltage orrectified ac output voltage available to one phase instead of two phasesin series, such that the available voltage to the machine phase becomesdoubled compared to the related art system of FIG. 1. In the system ofFIG. 1, two phase windings are in series and their currents are equalbut the applied voltage to each phase winding from the dc supply voltageis half the dc supply voltage. The subject matter disclosed hereinprovides a unique way of applying the entire dc supply voltage to eachphase of the machine, without an additional power converter, such as aboost converter.

FIG. 3 illustrates an inverter controlled PMBDC motor drive system.Components of the drive system illustrated in FIG. 3 that are common tothose of the drive system of FIG. 1 are identified with the samereference characters in each drawing.

A dc supply voltage 100, obtained either from a rectifier that isconnected to a utility supply or from a battery, is provided to aninverter 110. A filtering capacitor C₁ is connected in parallel with dcsupply voltage 100, via an upper voltage rail 101 and an intermediatevoltage rail 104, to smooth the voltage variations of dc supply voltage100. A capacitor C2 is electrically connected between intermediatevoltage rail 104 and a lower voltage rail 103. The dc voltage existingacross capacitor C₁ is expressed as V_(s1), and that existing acrosscapacitor C₂ is expressed as V_(s2). Ideally, V_(s1) is the same voltageas supply dc voltage 100.

Inverter 110 is a three-phase H bridge inverter and is electricallyconnected in parallel with the series connection of capacitors C₁ andC₂, via voltage rails 101 and 103. Inverter 110 has two transistors ineach of three phase legs. Phase leg A includes transistors T₁ and T₄that are electrically connected in series. A diode D₁ is connected inparallel with transistor T₁, such that the cathode of diode D₁ iselectrically connected to transistor T₁'s collector and the anode ofdiode D₁ is electrically connected to the emitter of transistor T₁. Adiode D₄ is similarly connected in parallel with transistor T₄. Phaseleg B includes transistors T₃ and T₆ that are electrically connected inseries, and phase leg C includes transistors T₅ and T₂ that areelectrically connected in series. Each of transistors T₂, T₃, T₅, and T₆has a corresponding diode D₂, D₃, D₅, and D₆ electrically connected inparallel with the respective transistor, in the same way that diode D₁is connected to transistor T₁.

The current conduction in each of transistors T₁-T₆ and diodes D₁-D₆ isindicated by an arrow in FIG. 1. Each of transistors T₁-T₆ conductscurrent from collector to emitter, and each of diodes D₁-D₆ conductscurrent from anode to cathode.

The collectors of transistors T₁, T₃, and T₅ and cathodes of diodes D₁,D₃, and D₅ are electrically connected to upper voltage rail 101. Theemitters of transistors T₂, T₄, and T₆ and anodes of diodes D_(2,) D₄,and D₆ are electrically connected to lower voltage rail 103. The gatesof transistors T₁-T₆ are individually controlled to govern the flow ofcurrent through the respective transistor.

A PMBDC motor 105 has three phase stator windings with phases A, B, andC. Phase leg A controls the current flow through the windings of phaseA, phase leg B controls the current flow through the windings of phaseB, and phase leg C controls the current flow through the windings ofphase C. One end of the stator winding for phase A is electricallyconnected to neutral N and the other end is electrically connected tothe emitter of transistor T₁, the collector of transistor T₄, the anodeof diode D₁, and the cathode of diode D₄. Similarly, one end of thestator winding for phase B is electrically connected to neutral N andthe other end is electrically connected to the emitter of transistor T₃,the collector of transistor T₆, the anode of diode D₃, and the cathodeof diode D₆. And similarly, one end of the stator winding for phase C iselectrically connected to neutral N and the other end is electricallyconnected to the emitter of transistor T₅, the collector of transistorT₂, the anode of diode D₅, and the cathode of diode D₂. Neutral N iselectrically connected to intermediate voltage rail 104.

Consider the operation of phase A as characteristic of that for each ofphases A-C. Turning on transistor T₁ applies voltage V_(s1) to phase Awinding, causing a positive current to flow from upper voltage rail 101through transistor T1 and phase winding A to intermediate voltage rail104. If the induced electromotive force (emf) across phase winding A ispositive, positive power is supplied to phase winding A, resulting inpositive torque and positive speed (CW). As positive power is applied tophase winding A, energy is transferred from dc supply voltage 100 tomotor 105. Motor 105's forward motoring operation is defined by a firstquadrant I of the torque versus speed characteristics illustrated inFIG. 5.

The four quadrant operation of the PMBDC motor drive is shown in FIG. 5.The x axis of the illustrated coordinate system denotes rotor speed, andthe y axis denotes the applied torque. Particularly, positive x axis 200corresponds to positive, forward, or clockwise rotation in speed whilethe negative x axis 210 corresponds to negative, reverse, orcounterclockwise rotation in speed. Positive torque corresponds topositive y axis 205, and negative torque corresponds to negative y axis215.

When transistor T₁ is turned off to control the current in phase A,energy stored by phase winding A is conveyed by diode D₄ to capacitorC2, which is charged to a voltage of V_(s2). During this time, energy istransferred from motor 105 to capacitor C₂. The applied voltage acrossphase winding A is −V_(s2) at this time and the torque is positive, asthe current and speed are still positive; therefore, the power output ispositive. This operation also corresponds to quadrant I in FIG. 5.

Suppose the speed of motor 105 is positive and transistor T₄ is turnedon to circulate a current from C₂ to phase winding A in a negativedirection while the induced emf is positive (i.e., with the positivepolarity of the induced emf existing at the collector of transistor T₄and the negative polarity existing at neutral N). This action will builda large current in phase winding A and, in order to keep within safecurrent limits, transistor T₄ will have to be turned off. Then, thecurrent in phase winding A will find a path through diode D₁ to chargecapacitor C₁, thereby transferring energy from motor 105 to dc supplysource 100. During this time, the induced emf is positive and current isnegative in phase winding A, with the result that motor 105 produces anegative torque (i.e., generative torque and hence negative power). Thisoperation corresponds to quadrant IV of FIG. 5. Hence, for positive orforward rotation of the rotor, quadrants I and IV correspond to forwardmotoring and forward regeneration.

Likewise quadrant I is replaced by quadrant III and quadrant IV replacedby quadrant II for reverse motoring and reverse regeneration,respectively. For brevity, those operations are not elaborated here asthey can be deduced and derived from the foregoing understanding ofquadrant I and IV operations using the inverter. This description ofoperation proves that the motor drive system of FIG. 3 provides full dcbus voltage V_(s1) to each phase winding during its positive cycles.Also the description proves that the motor drive system of FIG. 3 lendsitself to four quadrant operation.

In addition, the motor drive system of FIG. 3 makes possible full dc busvoltage application to the machine phases while creating an additionalvoltage source for higher degree of freedom in operation. This is madepossible by cycling energy, from the turn off of currents in thepositive half cycles of the phases, to charge capacitor C₂, resulting indc voltage source V_(s2). DC voltage source V_(s2) is applied during thenegative half cycle operation of the machine phases with the fullmagnitude of V_(s2) being applied to the phases of motor 105. Themagnitude of V_(s2) is comparable to V_(s1).

The voltage available for operation in quadrants I and IV is V_(s1) andin quadrants of II and III is V_(s2). The fact that V_(s2) is derivedfrom V_(s1) through the motor drive system of FIG. 3 during the positivehalf-cycle turn off of the phases makes it a somewhat dependent source.Source voltage V_(s2) can be controlled by the inverter operation andits application to machine phase windings increases the phaseroot-mean-square (rms) (or effective) voltage with the result that thetorque and power harvested out of the machine is increased. The phasevoltage of the motor drive system of FIG. 3 is twice that of the motordrive system of FIG. 1.

Only one dc input supply voltage, V_(s1), is available for the motordrive system of FIG. 3. With that, another voltage source, V_(s2), foroperating motor 105 is generated using capacitor C₂ and the bottomdiodes of inverter 110. In the process, a separate buck or boost orbuck-boost circuit is not required to create the additional voltagesource. Therefore, the motor drive system of FIG. 3 does not requireadditional devices to achieve higher phase voltages, but uses only thedevices that exist in inverter 110.

Further, no additional charging of capacitor C₂ takes place withoutaffecting, by controlling the magnitude of the current in the machine,the current control using pulse width modulation and resulting torquecontrol. Two voltage sources, V_(s1) and V_(s2), being available tocontrol and generate torque in PMBDC motor 105 endows many choices inthe control and maximization of efficiency in motor 105. The motor drivesystem of FIG. 3 supports the operation of the PMBDC machine with mostlyunidirectional current, using only source voltage V_(s1) in the positivecycle, or partly bidirectional current control using both sources,V_(s1) for positive half cycle currents and V_(s2) for negative halfcycle currents, as illustrated in FIG. 4.

With the drive control system of FIG. 3, the phase voltage (i.e., thevoltage applied across each phase winding) is the entire supply voltageV_(s1). By contrast, the motor drive system of FIG. 1 has two phasewindings in series such that each phase is supplied with half the inputvoltage (i.e., 0.5V_(s1)). Implications of this feature are describedbelow.

Consider the PMBDC machine drive system of FIG. 1 in relation to that ofFIG. 3. The phase current requirement for an input power of V_(s1)I iscalculated as follows and given in Table 1.

TABLE 1 Terms FIG. 1 System FIG. 3 System Current BidirectionalBidirectional Phase voltage V_(s1)/2 V_(s1) Phase current to be derivedbased on power V_(s1)I $\frac{V_{s\; 1}{I/2}}{V_{s\; 1}/2} = I$$\frac{V_{s\; 1}I}{V_{s\; 1}} = {I({unidirectional})}$${\frac{1}{2}\frac{V_{s\; 1}I}{V_{s\; 1}}} = {\frac{I}{2}({bidirectional})}$

For only unidirectional current in the machine phases of the system ofFIG. 3, the current requirement is the same as for the system of FIG. 1.For the unidirectional current operation of the system illustrated inFIG. 3, which operation is illustrated in FIG. 6, only positivehalf-cycle currents flow in the phase windings of motor 105. Thepositive-cycle current operation of any phase lasts only for 120°, whichis one third of the full cycle of a phase voltage or current having 360electrical degrees. Therefore, the phase currents exist for only ⅓ ofthe time and have a duty cycle d of ⅓. The rms value of a constantcurrent I_(p) having a duty cycle d is I/(d)^(1/2). Therefore, the rmscurrent through a phase winding during a duty cycle of ⅓I_(p)/(3)^(1/2).

The motor drive system of FIG. 1 has phase conduction for both positiveand negative half cycles for electrical 120° in each of these cycles.Therefore, the phase currents are present for electrical 240° whichconstitutes ⅔ of the entire phase cycle. Thus, the duty cycle of thephase currents is ⅔ and the rms current is calculated from currentmagnitude divided by the square root of the duty cycle, which isexpressed as I_(p)/(⅔)^(1/2).

The difference in the rms phase winding currents of the systems of FIGS.1 and 3 are also reflected in the average phase winding currents forthese two systems. Thus, the current rating of the motor drive systemillustrated in FIG. 3 is lower than that for the system of FIG. 1.

The motor drive system of FIG. 1 cannot provide unidirectional currentin the phase windings of motor 105. The reason for this is that thephase windings are connected in series and for a positive current toflow in one phase winding the flowing current must come from theseries-connected phase winding. But the current flowing into the onephase winding is positive while the current flowing out of the otherphase is negative. Because the current in one phase winding must beequal and opposite to the current in the other phase winding of theseries connected phase windings, it is impossible to have unidirectionalcurrents. Therefore, unidirectional currents cannot be applied to thephase winding of the motor drive system illustrated in FIG. 1.

As illustrated in FIG. 6, the induced emf of a phase is 2E_(p) for thesystem of FIG. 3, whereas the induced emf for a phase of the systemillustrated by FIG. 1 is E_(p), as illustrated in FIG. 2. Since twophase windings are in series in the system of FIG. 1, the total inducedemf facing the supply input voltage is twice the induced emf per phase,which then is equal to 2E_(p). For the system illustrated in FIG. 3, thesupply dc input voltage faces only one induced emf per phase at anytime. Assuming the supply voltage is the same for the systems of FIGS. 1and 3, the same dc input voltage supply should be able to support thesame induced emf facing it, which is 2E_(p). Thus, the number of turnsfor a phase of the system illustrated by FIG. 3 may be doubled withrespect to those of the system of FIG. 1, so as to achieve the sameamount of induced emf facing the voltage supply. With equal currents ofI_(p) being drawn for each phase in both the systems of FIGS. 1 and 3,the air gap power is the same, as illustrated in FIGS. 2 and 6, andthere is no difference in power output or torque for the two systems

For bipolar current in the machine phases of the system illustrated inFIG. 3, the current in the phases is 0.5I_(p), which is 50% less thanthat within the system of FIG. 1. This advantage in current rating forthe system of FIG. 3 is due to the doubling of the phase voltageavailable to the phase windings. The doubling of the available phasevoltage can also be used to extend the operational speed range of amotor, as described later.

Assume that unidirectional current plays a major role in torquegeneration for the system of FIG. 3 and that there are no bidirectionalcurrents. In this case, the number of winding tunes in the system ofFIG. 3 has to be twice that of the system of FIG. 1 to achieve the sameinduced emf. Further assume that the copper volume and cost is matchedin both systems. From these assumptions, the information within Table 2may be derived for the systems of FIGS. 1 and 3.

TABLE 2 Categories FIG. 1 System FIG. 3 System 1. Number of turns perphase N_(s) 2N_(s) 2. Wire cross-section a_(c) $\frac{a_{c}}{2}$ 3.Copper volume in machine αN_(s)a_(c) Same 4. Torque constant Nm/A perK_(t) 2K_(t) phase 5. Supply voltage/phase $\frac{V_{s\; 1}}{2}$ V_(s1)6. Power/phase $\frac{V_{s\; 1}}{2}I_{p}$ $V_{s\; 1}\frac{I_{p}}{2}$ 7.Total power V_(s1)I_(p) V_(s1)[I_(p)] 8. Phase current for equal I_(p)I_(p) (unidirectional) power 0.5I_(p) (bidirectional) 9. Phaseresistance in machine R_(s) 4R_(s)

For a bipolar drive, the phase currents are I/2 for the systemillustrated by FIG. 3. The subscript p is deleted hereafter, forsimplicity, from current I. The total stator resistive loss for any twoconducting phase windings, for example phases A and B, with phase Acurrent I_(a) and phase B current I_(b), is:

$\begin{matrix}{P_{cu} = {{( {I_{a}^{2} + I_{b}^{2}} )4R_{s}} = {{( {\frac{I^{2}}{4} + \frac{I^{2}}{4}} )4R_{s}} = {2I^{2}R_{s}}}}} & (13)\end{matrix}$

And this resistive loss is the same as that for the system illustratedby FIG. 1. Thus, for bipolar operation, the systems of FIGS. 1 and 3have the same resistive loss. However, the inverter losses are also animportant consideration for improving the efficiency of a machine.

Consider the transistor conduction loss in the inverter. The switchinglosses are usually smaller compared to the conduction loss and, hence,are not considered.

Let V_(t) designate the conduction voltage drop in each transistor ofthe system of FIG. 3, and let each phase transistor carry only half therated current to produce the equivalent power produced by the system ofFIG. 1. Then, the conduction loss P_(t) for the two conducting phases atany given time in the system of FIG. 3 is:

$\begin{matrix}{P_{t} = {{V_{t}\lbrack {\frac{I}{2} + \frac{I}{2}} \rbrack} = {V_{t}I}}} & (14)\end{matrix}$

In the system illustrated by FIG. 1, two transistors conduct to carrythe same load current I at any given time, resulting in two transistorconduction losses P_(tc) given by:P _(tc) =V _(t) I×2=2V _(t) I  (15)

Therefore, the ratio of conduction loss between the systems of FIGS. 1and 3 is:

$\begin{matrix}{\frac{P_{t}}{P_{tc}} = {\frac{V_{t}I}{2V_{t}I} = \frac{1}{2}}} & (16)\end{matrix}$

Thus, the system of FIG. 3 reduces the conduction power loss by 50% withrespect to that of the system of FIG. 1. This is on the order of 0.5 kWto 1 kW for an electric vehicle hybrid drive. And due to the lowerconduction loss, the cooling requirement for the inverter is reduced,resulting in higher thermal robustness and reliability for the inverter.

The above description assumes that the phase voltage for the system ofFIG. 3 is doubled with respect to that of the system of FIG. 1 and thateach system provides the same base speed and base torque. An attractiveparametric study is that in which the phase voltage at base speed forthe system of FIG. 3 is something other than twice, such as less thantwice, that of the system of FIG. 1.

Let the applied phase voltage for the system of FIG. 3 be expressed as:v _(an) =kv _(dc) =k(2V _(a))=(2k)V _(a); 0<k≦1  (17)

where V_(a) is the applied phase voltage for the system of FIG. 1. Atbase speed, the applied phase voltage is equal to V_(dc)/2, where V_(dc)is the dc supply voltage input to the inverter. Factor k varies theapplied voltage from 2V_(a) to 2 kV_(a) or kv_(dc) through the inverter.Accordingly, the system of FIG. 3 has to have a base voltage (i.e., atbase speed) of 2 kV_(a) and, hence, the number of winding turns has tobe multiplied by 2k as seen from emf equation (18).N=(2k)N _(s)  (18)

Keeping the copper volume constant, for achieving a constant copper costbetween the systems of FIGS. 1 and 3, results in a conductor crosssection a_(c) of:

$\begin{matrix}{a_{c} = \frac{a}{2k}} & (19)\end{matrix}$

where a is the cross section of the conductor in the system of FIG. 1.The resistance of the machine of the system of FIG. 3 becomes:R=(4k ²)R _(s)  (20)

where R_(s) is the per phase resistance of the system of FIG. 1. Thecopper loss in the system of FIG. 1, in which a factor of I isintroduced, is:P _(cu)=(I _(a) ² +I _(b) ²)R=(I _(a) ² +I _(b) ²)(4k ²)R _(s)  (21)

The ratio of copper losses between the systems of FIGS. 1 and 3 isexpressed as;

$\begin{matrix}{\frac{P_{{cu}{({new})}}}{P_{{cu}{({conventional})}}} = \frac{( {I_{a}^{2} + I_{b}^{2}} )( {4k^{2}} )R_{s}}{( {I_{ac}^{2} + I_{bc}^{2}} )R_{s}}} & (22)\end{matrix}$But I _(ac) =I _(bc) =I (in system of FIG. 1)  (23)

$\begin{matrix}{{{{And}\mspace{14mu} I_{a}} + I_{b}} = {\frac{I}{k}\mspace{14mu}( {{in}\mspace{14mu}{system}\mspace{14mu}{of}\mspace{14mu}{{FIG}.\mspace{14mu} 3}} )}} & (24)\end{matrix}$Letting I _(a) =I _(b) in the above equation gives:  (25)

$\begin{matrix}{{I_{a} + I_{b}} = {{2\; I_{a}} = \frac{I}{k}}} & (26)\end{matrix}$

which gives way to:

$\begin{matrix}{I_{a} = \frac{I}{2\; k}} & (27)\end{matrix}$

The resistive loss in the system of FIG. 3 may be expressed as:

$\begin{matrix}{P_{{cu}{(n)}} = {{\lbrack \frac{I}{2k} \rbrack^{2}2*4k^{2}R_{s}} = {2I^{2}R_{s}}}} & (28)\end{matrix}$

And that in the system of FIG. 1 may be expressed as:P _(cu(c))=2I ² R _(s)  (29)

Thus, the systems of FIGS. 1 and 3 have the same copper loss under theabove-described circumstance.

The ratio of inverter conduction power losses is:

$\begin{matrix}{{P_{tn} = {V_{t}{I/k}}}{P_{tc} = {2V_{t}I}}{\frac{P_{tn}}{P_{tc}} = {\frac{V_{t}{I/k}}{2V_{t}I} = \frac{1}{2k}}}} & (30)\end{matrix}$

where P_(tn) is the inverter conduction losses with the system of FIG. 3and P_(tc) is the inverter conduction losses with the system of FIG. 1.Some sample calculations are supplied in Table 3 to provide a feel forthe implications of varying k, with respect to the number of turns perphase winding, resistance per phase, phase current, and ratio ofconduction losses for the systems of FIGS. 1 and 3. N_(s) is the numberof turns per phase, R_(s) is the resistance per phase, and I is the perphase current in the system of FIG. 1.

TABLE 3 Number of Resistance I_(a), Phase k turns per phase per phasecurrent P_(tn)/P_(tc) 0.6 1.2 N_(s) 1.44 R_(s) 0.833 I 0.835 0.707 1.414N_(s)    2 R_(s) 0.707 I 0.707 0.8 1.6 N_(s) 2.56 R_(s) 0.625 I 0.625

Based on these numerical examples, the following inferences can be made:

1. As k increases, P_(tn) decreases inversely.

2. Reserve voltage available to a machine phase is(v_(dc)−kv_(dc))=(1−k)v_(dc), and this increases with decreasing k.

3. The reserve voltage can be used to obtain a speed of operation thatis greater than that of the base speed (at which machine delivers ratedor base torque).

4. The new base speed for the system of FIG. 3 is 1/k in per unit. Forexample:

TABLE 4 k New base speed, p.u. 0.6 1.667 0.707 1.414 0.8 1.25

Table 4 exemplifies that a very large extension of base speed, at whichbase torque is obtained, may be achieved by changing the value of k in adesign. In Table 4, p.u. is the normalized speed, which is nondimensional and limited to low values, such as 1 or 2 or 3. For example,speed in p.u. is equal to the actual speed in revolutions per minute(rpm) divided by base, which is usually the rated speed. Suppose thatthe actual speed is 1000 rpm and base speed is 2000 rpm, then the speedin p.u. is equal to 1000/2000=0.5 p.u. This allows scaling to be doneregardless of the variations in many variables such as in voltages,currents, torque, and power output by using base values for each ofthese variables to obtain manageable numbers of p.u. values.

5. Base speed extension also allows for higher range of flux-weakeningoperation, with the result that the system of FIG. 3 provides anexcellent fit for electric vehicle or hybrid motor drive.

The system of FIG. 3 enables the operation of each phase independentlyof the other phases. For example, if the phases are given onlyunidirectional currents, then the total power and torque can beharvested from only one phase at any time, which has consequences onresistive losses, transistor conduction losses, and total losses. Forexample:Resistive losses=4R _(s) I ²  (31)Transistor conduction loss=V _(t) I  (32)P _(ut)=total losses using only unidirectional currents=V _(t) I+4R _(s)I ²  (33)Likewise:P _(et)=total drive losses of the system of FIG. 1=2V _(t) I−2R _(s) I²  (34)Equating losses for the systems of FIGS. 1 and 3 leads to:2V _(t) I=2R _(s) I ²  (35)which, in turn leads to the following condition:

$\begin{matrix}{I = \frac{V_{t}}{2R_{s}}} & (36)\end{matrix}$orV _(t)=2R _(s) I  (37)Two scenarios arise in regard to power losses:(i) If V_(t)>2R_(s)I, then the system of FIG. 3 provides lower powerlosses.(ii) If V_(t)<2R_(s)I, then the system of FIG. 1 provides lower losses.

For example, in electric vehicle drives, V_(t)≧2R_(s)I and, thus, thesystem of FIG. 3 will be better than that of FIG. 1. Even ifV_(t)≦2R_(s)I, the deficit in losses for the system of FIG. 3 can bewiped out by using the energy in C₂ to make a bidirectional drive, whichwill reduce the positive current to provide high efficiency operation.Further, the commutation of positive current is faster in the system ofFIG. 3 because it charges C₂ directly from the phase current windingwithout involving another phase during its commutation. In addition, thesystem of FIG. 3 provides greater fault tolerance for a failed machinephase or inverter transistor.

It may be stated that mostly single-phase operation with a small amountof energy harvested with another phase excitation is an important modeof operation for the system of FIG. 3. That does not mean other modes ofoperation are excluded; other modes will play a role depending on theapplication of the motor drive system. The control principle for this isderived here where bipolar currents are used in machine phases but onlysmall negative currents, as opposed to large positive currents, areemployed. For illustration of this control strategy, consider the motordrive having a phase sequence of a,b,c. The conventional operation withbipolar currents is shown in Ref [R Krishnan, “Electric Motor Drives”,Prentice Hall, 2001, pp. 523]. Consider a time instant when phase Acurrent is positive and phase B current has to be negative. Let thephase currents be scaled up or down by a fact or of a and b for phases Aand B, respectively, from their rated or base value of I and be denotedas:

A phase current I_(a)=aI, and

B phase current I_(b)=bI, where I is the rated current.

Then:T _(e)=(a+b),p.u. I=1 p.u.  (38)P _(cu)=(a ² +b ²)4R _(s) I ²  (39)

Desired torque equals the rated torque, which may be defined as 1p.u.(where p.u. is the per unit used as normalized non-dimensional unit and1 p.u. indicates 100% or rated value of a variable, be it current,voltage, torque, power, etc.).

$\begin{matrix}{{{\therefore{a + b}} = 1};{a = {1 - b}};{b = {1 - a}}} & (40) \\\begin{matrix}{P_{cn} = {\lbrack {a^{2} + ( {1 - a} )^{2}} \rbrack 4R_{s}I^{2}}} \\{= {\lbrack {{2a^{2}} - {2a} + 1} \rbrack 4R_{s}I^{2}}}\end{matrix} & (41)\end{matrix}$

Equate this to conventional operation P_(cu) loss as

$\begin{matrix}{{{\lbrack {{2a^{2}} - {2a} + 1} \rbrack 4R_{s}I^{2}} = {2R_{s}I^{2}}}{{{{2a^{2}} - {2a} + \frac{1}{2}} = {{0\therefore a} = \frac{1}{2}}};{b = \frac{1}{2}}}} & (42)\end{matrix}$

Include device conduction loss; then the equations become for the systemof FIG. 3:a+b=1, P _(in)=(a ² +b ²)4R _(s) I ²+(a+b)v _(t) I  (43)

Where v_(t) is the transistor conduction voltage drop in the inverter.And for the system of FIG. 1, the total loss is:P _(tc)=2R _(s) I ²+2v _(t) I  (44)

If a and b are fractions and do not add up to 1, different results arepossible. Both cases will be examined (i.e., a+b=1, and a+b<1) in thefollowing. Consider:

$\begin{matrix}\begin{matrix}{{P_{tn} = \frac{{( {a^{2} + b^{2}} )4R_{s}I^{2}} + {( {a + b} )v_{t}I}}{{2R_{s}I^{2}} + {2v_{t}I}}},\{ \begin{matrix}{I_{a} = {aI}} \\{I_{b} = {bI}}\end{matrix} } \\{= \frac{{( {a^{2} + b^{2}} )4R_{s}I} + {( {a + b} )v_{t}}}{{2R_{s}I} + {2v_{t}}}} \\{= \frac{{( {a^{2} + b^{2}} )4\frac{R_{s}I}{v_{t}}} + ( {a + b} )}{{2\frac{R_{s}I}{v_{t}}} + 2}}\end{matrix} & (45)\end{matrix}$The ratio of the total loss P_(r) between the systems of FIGS. 3 and 1is given by:

$\begin{matrix}{P_{r} = {\frac{P_{tn}}{P_{tc}} = \frac{{4{R_{sn}( {a^{2} + b^{2}} )}} + ( {a + b} )}{2( {R_{sn} + 1} )}}} & (46)\end{matrix}$where the normalized stator resistance R_(sn) is defined as:

$\begin{matrix}{{R_{sn} = \frac{R_{s}I}{v_{t}}},{p.u.}} & (47)\end{matrix}$

Case (i): a+b=1 and P_(r)=1

$\begin{matrix}{P_{r} = {1 = \frac{{4{R_{sn}( {a^{2} + \{ {1 - a} \}^{2}} )}} + 1}{2( {R_{sn} + 1} )}}} & (48) \\\begin{matrix}{{{2R_{sn}} + 2} = {{4R_{sn}\lfloor {{2a^{2}} - {2a} + 1} \rfloor} + 1}} \\{= {{8R_{sn}a^{2}} - {8R_{sn}a} + {4R_{sn}} + 1}}\end{matrix} & (49)\end{matrix}$leading to:(8R _(sn))a ²−(8R _(sn))a+2R _(sn)−1=0  (50)Solving for a:

$\begin{matrix}\begin{matrix}{a = \frac{{8R_{sn}} \pm \sqrt{{64R_{sn}^{2}} - {32{R_{sn}( {{2R_{sn}} - 1} )}}}}{16R_{sn}}} \\{= {0.5 \pm {0.3535\sqrt{\frac{1}{R_{sn}}}}}}\end{matrix} & (51)\end{matrix}$

Phase current for the positive conducting phase, identified as the firstphase in FIG. 4, versus I for various R_(sn), ranging from 0.5 to 3p.u., is shown in FIG. 7. FIG. 7 illustrates the current in positivephase, normalized units. From the value of a, the value of b can beextracted as b=1−a, and this is the normalized current in negativephase. The latter indicates the current to be supplied by C₂ (i.e.,auxiliary capacitor). FIG. 6 shows the operating points with theconstraints that the sum of two phase currents equal one p.u. and theloss in each of systems 1 and 3 is the same. As may be determined fromFIG. 7, for R_(sn)=2.1 or higher values and for a<0.7, the losses in thesystem of FIG. 3 are attractive compared to those of the system of FIG.1.

Case(ii): a+b<1 means less than rated current being supplied, say forpartial loads.

In this case:

$\begin{matrix}{P_{r} = \frac{{4{R_{sn}( {a^{2} + b^{2}} )}} + ( {a + b} )}{2\{ {{R_{sn}( {a + b} )}^{2} + ( {a + b} )} }} & (52)\end{matrix}$If a+b=m, then b=(m−a)  (53)

$\begin{matrix}{{\therefore P_{r}} = \frac{{4R_{sn}\{ {a^{2} + a^{2} - {2{am}} + m^{2}} \}} + m}{2\lbrack {{R_{sn}m^{2}} + m} \rbrack}} & (54)\end{matrix}$

Given the value of m, the value of a can be found and then the value ofb for a given P_(r) (assume it is equal to 1, i.e., P_(r)=1). Note thatthe torque is assumed to be equal for both systems of FIGS. 1 and 3.

Case(iii): a=b, but a+b≦1.

$\begin{matrix}{P_{r} = {\frac{{4R_{sn}\lfloor {2a^{2}} \rfloor} + {2a}}{2\lbrack {{4a^{2}R_{sn}} + {2a}} \rbrack} = {\frac{{4R_{sn}a^{2}} + a}{{4a^{2}R_{sn}} + {2a}} = \frac{{4{aR}_{sn}} + 1}{2\lbrack {{2{aR}_{sn}} + 1} \rbrack}}}} & (55)\end{matrix}$

Under the constraints described above, the system of FIG. 3 has lowerloss than that of the system of FIG. 1, as the numerator of the aboveequation is smaller than the denominator. Such is advantageous for apartial load condition, which is what most drive applications encounterin practice.

Consider the system of FIG. 1 and assume that one of the machine phasewindings has an open-circuit fault, in such a case, current in the othertwo phase windings cannot be controlled as required with electrical 120°in both their positive and negative half cycles. Contrast this with thesystem of FIG. 3 in which only the phase winding having an open-circuitfault is deprived of current. The remaining phases will get theirrespective currents, as shown in FIG. 4, without any change as comparedto a pre-fault condition. Similar reasoning can be applied toshort-circuit fault cases in the machines, with the same advantage beingmaintained in the system of FIG. 3. Similar reasoning is extendable toinverter transistor failures and the availability of currents to phasesthat are unaffected by the fault.

Although the currents in the above discussion have been considered asideal currents, ideal currents are not possible to realize, but may beclosely realized by resorting to pulse width modulation (PWM) control ofthe inverter transistors. The control of transistor inverters to shapecurrents is well known in text books, one of which is R. Krishnan,“Electric Motor Drives”, Prentice Hall, 2001. Those knowledgeable in theart are familiar with PWM control of a transistor inverter.

Variations of the invention exist, such as provisioning the dc supplyvoltage from a battery or connecting an ac-rectified dc supply across C₂instead of C₁ within the system of FIG. 3. The change in the supplyinput points supports negative half-cycle based unidirectional currentcontrol, without affecting bidirectional current control. The change inthe supply input points does not affect the torque or power output.

Consider a case in which the top capacitor C₁ in the system of FIG. 3 isconnected to a dc supply voltage, obtained from a rectified ac supply,and a battery is connected across C₂ with the illustrated polarity. Ifthe ac supply fails, the dc input voltage to the capacitor C₁ is lost,but C₁ can be charged from the battery across C₂ by using the bottomtransistors, machine phase windings, and respective top diodes of theinverter. The battery can be kept charged when the ac supply isavailable, which is then rectified and fed to charge C₁. Then, thecharge stored by C₁ can be transferred to C₂ using the machine phasewindings, top transistors, and bottom diodes of the inverter. Therefore,this system provides an uninterrupted power supply to run the PMBDCmotor drive when one of the power sources, either from a rectified acsupply or from a battery is absent. This arrangement of having thebattery across C₂ allows the battery to be charged through thecontrolled flow of current in the machine windings. Therefore, batterycharging is accomplished without additional components, such as atransistor, an inductor, and a circuit breaker that usually are requiredin a separate charger such as buck or boost charger.

FIG. 8 illustrates such an uninterruptable power supply for a powerconverter. FIG. 8 illustrates the same system as FIG. 3, with theexception that a battery or dc supply is connected in parallel withcapacitor C₂. With this arrangement, the uninterruptable operation ofthe system illustrated in FIG. 3 may be achieved as described in thepreceding paragraph.

By contrast to the system of FIG. 1, the system of FIG. 3 provides:

1. higher reliability and fault tolerance;

2. higher speed of operation;

3. lower conduction losses in an inverter;

4. higher efficiency in the machine; and

5. higher efficiency operation.

Moreover, a split phase dc is created without an additional buck orboost or buck-boost converter at the front-end. And four quadrantoperation with a higher torque-speed envelope for transient operationand steady-state operation is achieved. Other benefits of the system ofFIG. 3 include:

1. full dc input voltage to the phase winding;

2. unidirectional current operation;

3. four quadrant operation;

4. bidirectional or bipolar current operation;

5. flexible operation to vary the magnitude of currents and voltagesacross the machine windings using pulse width modulation or other knownmeans;

6. minimization of total losses that include stator resistive losses andtransistor conduction losses;

7. extension of base speed by as much as two times with full rated orbase torque;

8. extension of base speed can be varied from 1 to 2 at least;

9. fault tolerance is high in both the machine as well as in theinverter;

10. uninterrupted operation of a PMBDC motor drive when a power sourcefails; and

11. controlled current charging of a battery from a rectified ac supplywithout additional components.

The foregoing has been a detailed description of possible embodiments ofthe invention. Other embodiments of the invention will be apparent tothose skilled in the art from consideration of the specification andpractice of the invention. Accordingly, it is intended that thisspecification and its disclosed embodiments be considered as exemplaryonly, with a true scope and spirit of the invention being indicated bythe following claims.

What is claimed is:
 1. A motor drive system comprising: a firstcapacitive element that stores energy to provide a first direct current(dc) voltage supply; a second capacitive element that stores energy toprovide a second dc voltage supply; a first electrically conductiveswitch for electrically connecting in series with a winding of a singlemotor phase and the first capacitive element to form a first seriescircuit when the first switch conducts current; a second electricallyconductive switch for electrically connecting in series with the motorphase winding and the second capacitive element to form a second seriescircuit when the second switch conducts current; a dc supplyelectrically connected in parallel with one of the first and secondcapacitive elements, such that substantially an entire voltage of the dcsupply is applied across the one of the first and second capacitiveelements; and a battery electrically connected in parallel with the oneof the first and second capacitive elements that is not electricallyconnected in parallel with the dc supply, such that substantially anentire voltage of the battery is applied across the one of the first andsecond capacitive elements that is not electrically connected inparallel with the dc supply, wherein: the entire voltage of the firstcapacitive element is applied across the motor phase winding, less thevoltage drop across the first switch and the voltage drop resulting fromparasitic resistance in the first series circuit, when: (1) the secondswitch does not conduct current and (2) the second capacitive element isneither being charged nor discharged, and the entire voltage of thesecond capacitive element is applied across the motor phase winding,less the voltage drop across the second switch and the voltage dropresulting from parasitic resistance in the second series circuit, when:(1) the first switch does not conduct current and (2) the firstcapacitive element is neither being charged nor discharged.
 2. The motordrive system of claim 1, wherein: the first series circuit passescurrent in a first polarity through the motor phase winding when thefirst switch conducts current, the second switch does not conductcurrent, and the second capacitive element is neither being charged nordischarged, and the second series circuit passes current in a secondpolarity, opposite to the first polarity, through the motor phasewinding when the second switch conducts current, the first switch doesnot conduct current, and the first capacitive element is neither beingcharged nor discharged.
 3. The motor drive system of claim 1, furthercomprising: an Nth electrically conductive switch, N being an integergreater than 1, for electrically connecting in series with an Nth motorphase winding and the first capacitive element to form an Nth seriescircuit when the Nth switch conducts current; and an Nth+1 electricallyconductive switch for electrically connecting in series with the Nthmotor phase winding and the second capacitive element to form an Nth+1series circuit when the Nth+1 switch conducts current, wherein: theentire voltage of the first capacitive element is applied across the Nthmotor phase winding, less the voltage drop across the Nth switch and thevoltage drop resulting from parasitic resistance in the Nth seriescircuit, when: (1) the Nth+1 switch does not conduct current and (2) thesecond capacitive element is neither being charged nor discharged, andthe entire voltage of the second capacitive element is applied acrossthe Nth motor phase winding, less the voltage drop across the Nth+1switch and the voltage drop resulting from parasitic resistance in theNth+1 series circuit, when: (1) the Nth switch does not conduct currentand (2) the first capacitive element is neither being charged nordischarged.
 4. The motor drive system of claim 3, wherein: the Nthseries circuit passes current in a first polarity through the Nth motorphase winding when the Nth switch conducts current, the Nth+1 switchdoes not conduct current, and the second capacitive element is neitherbeing charged nor discharged, and the Nth+1 series circuit passescurrent in a second polarity, opposite to the first polarity, throughthe Nth motor phase winding when the Nth+1 switch conducts current, theNth switch does not conduct current, and the first capacitive element isneither being charged nor discharged.
 5. The motor drive system of claim3, wherein an H-bridge inverter comprises original first, second, Nth,and Nth+1 switches.
 6. The motor drive system of claim 1, furthercomprising: a first unidirectional current element that passes currentin only one direction and is electrically connected in parallel with thefirst switch; a second unidirectional current element that passescurrent in only one direction and is electrically connected in parallelwith the second switch, wherein: the first or second unidirectionalcurrent element discharges energy stored by the motor phase winding tothe first or second capacitive element, respectively, when both thefirst and second switches are not conducting current.
 7. A motor drivesystem comprising: a phase winding of a motor that applies electromotiveforce to a rotor of the motor; a first capacitive element that storesenergy to provide a first direct current (dc) voltage supply; a secondcapacitive element that stores energy to provide a second dc voltagesupply; a first electrically conductive switch for electricallyconnecting in series with the motor phase winding and the firstcapacitive element to form a first series circuit when the first switchconducts current; a second electrically conductive switch forelectrically connecting in series with the motor phase winding and thesecond capacitive element to form a second series circuit when thesecond switch conducts current; a dc supply electrically connected inparallel with one of the first and second capacitive elements, such thatsubstantially an entire voltage of the dc supply is applied across theone of the first and second capacitive elements; and a batteryelectrically connected in parallel with the one of the first and secondcapacitive elements that is not electrically connected in parallel withthe dc supply, such that substantially an entire voltage of the batteryis applied across the one of the first and second capacitive elementsthat is not electrically connected in parallel with the dc supply,wherein: the entire voltage of the first capacitive element is appliedacross the motor phase winding, less the voltage drop across the firstswitch and the voltage drop resulting from parasitic resistance in thefirst series circuit, when: (1) the second switch does not conductcurrent and (2) the second capacitive element is neither being chargednor discharged, and the entire voltage of the second capacitive elementis applied across the motor phase winding, less the voltage drop acrossthe second switch and the voltage drop resulting from parasiticresistance in the second series circuit, when: (1) the first switch doesnot conduct current and (2) the first capacitive element is neitherbeing charged nor discharged.
 8. The motor drive system of claim 7,wherein: the first series circuit passes current in a first polaritythrough the motor phase winding when the first switch conducts current,the second switch does not conduct current, and the second capacitiveelement is neither being charged nor discharged, and the second seriescircuit passes current in a second polarity, opposite to the firstpolarity, through the motor phase winding when the second switchconducts current, the first switch does not conduct current, and thefirst capacitive element is neither being charged nor discharged.
 9. Amotor drive system comprising: a first capacitive element that storesenergy to provide a first direct current (dc) voltage supply; a secondcapacitive element that stores energy to provide a second dc voltagesupply; first, second, third, and fourth electrically conductiveswitches each comprising an input current terminal and an output currentterminal; a dc supply electrically connected in parallel with one of thefirst and second capacitive elements, such that substantially an entirevoltage of the dc supply is applied across the one of the first andsecond capacitive elements; and a battery electrically connected inparallel with the one of the first and second capacitive elements thatis not electrically connected in parallel with the dc supply, such thatsubstantially an entire voltage of the battery is applied across the oneof the first and second capacitive elements that is not electricallyconnected in parallel with the dc supply, wherein: a first terminal ofthe first capacitive element is electrically connected directly to theinput current terminals of the first and second switches, a secondterminal of the first capacitive element is electrically connecteddirectly to a first terminal of the second capacitive element, a secondterminal of the second capacitive element is electrically connecteddirectly to the output current terminals of the third and fourthswitches, the output current terminal of the first switch iselectrically connected directly to the input current terminal of thethird switch, the output current terminal of the second switch iselectrically connected directly to the input current terminal of thefourth switch, a first terminal of a winding of a first motor phase iselectrically connected directly to the second terminal of the firstcapacitive element and the first terminal of the second capacitiveelement, a second terminal of the first motor phase winding iselectrically connected directly to the output current terminal of thefirst switch and the input current terminal of the third switch, a firstterminal of a winding of a second motor phase is electrically connecteddirectly to the second terminal of the first capacitive element and thefirst terminal of the second capacitive element, a second terminal ofthe second motor phase winding is electrically connected directly to theoutput current terminal of the second switch and the input currentterminal of the fourth switch.
 10. The motor drive system of claim 9,further comprising: fifth and sixth electrically conductive switcheseach comprising an input current terminal and an output currentterminal, wherein: the first terminal of the first capacitive element iselectrically connected directly to the input current terminal of thefifth switch, the second terminal of the second capacitive element iselectrically connected directly to the output current terminal of thesixth switch, the output current terminal of the fifth switch iselectrically connected directly to the input current terminal of thesixth switch, a first terminal of a winding of a third motor phase iselectrically connected directly to the second terminal of the firstcapacitive element and the first terminal of the second capacitiveelement, a second terminal of the third motor phase winding iselectrically connected directly to the output current terminal of thefifth switch and the input current terminal of the sixth switch.
 11. Amotor drive method comprising: charging a first capacitive element usinga direct current (dc) power supply electrically coupled in parallel withthe first capacitive element; wherein substantially an entire voltage ofthe dc power supply is applied across the first capacitive element;charging a second capacitive element using a battery electricallycoupled in parallel with the second capacitive element; wherein thesecond capacitive element is not electrically connected in parallel withthe dc supply and substantially an entire voltage of the battery isapplied across the second capacitive element; discharging energy storedby the first capacitive element into a single electrical phase of amotor, via a first electrically conductive switch, such that the entirevoltage across the first capacitive element is applied across the motorphase winding, less the voltage drop across the first switch and thevoltage drop resulting from parasitic resistance in a first seriescircuit comprising the first capacitive element, the first switch, andthe motor phase winding; and discharging energy stored by the motorphase winding into a second capacitive element, via a secondelectrically conductive switch, such that the entire voltage across themotor phase winding is applied across the second capacitive element,less the voltage drop across the second switch and the voltage dropresulting from parasitic resistance in a second series circuitcomprising the second capacitive element, the second switch, and themotor phase winding.